Results 1 to 8 of 8

Math Help - D'Alembert PDE

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    D'Alembert PDE

    Let L=\frac{\partial^2}{\partial t^2}-c^2\frac{\partial^2}{\partial x^2}

    with Lu=0 and Lv=0. In other words, let u_{tt}-c^2u_{xx}=0 and v_{tt}-c^2v_{xx}=0. Let w=u_tv_t+c^2u_xv_x. It can be shown that Lw=0.

    Prove that for a<x<b, t>0 and u=v=0 for x=a,b, t>0, that

    \frac{d}{dt}\int_a^b\frac{1}{2} w\;dx=0.

    I'm really stuck on this one. Any help would be much appreciated. Thanks!

    (By the way, this is from Fritz John, Partial Differential Equations, exercise 2.4.7b., p46.)

    EDIT: Thanks for the help! It is now solved, with the following proof:

    Quote Originally Posted by me
    Since u(a,t)=v(a,t)=u(b,t)=v(b,t)=0 for all t>0, we have u_t(a,t)=v_t(a,t)=u_t(b,t)=v_t(b,t)=0 for t>0. Recall also that u_{tt}=c^2u_{xx} and v_{tt}=c^2v_{xx}. These facts allow us to see that

    \frac{d}{dt}\int_a^b\frac{1}{2}(u_tv_t+c^2u_xv_x)d  x

    =\int_a^b\frac{1}{2}(u_{tt}v_t+u_tv_{tt}+c^2(u_{xt  }v_x+u_xv_{xt}))dx

    =\frac{c^2}{2}\int_a^b(u_{xx}v_t+u_xv_{xt}+u_{xt}v  _x+u_tv_{xx})dx

    =\frac{c^2}{2}\left[u_xv_t+u_tv_x\right]_a^b=0. \blacksquare
    Last edited by hatsoff; September 1st 2011 at 06:09 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41

    Re: D'Alembert PDE

    Have you seen the following:

    If u_{tt} = c^2 u_{xx} on a < x < b with u(a,t) = 0, u(b,t) = 0,

    if we define

    e(t) = \int_a^b u_t^2 + c^2 u_x^2 dx

    then

    \frac{de}{dt} = 0.

    Have you seen this proven?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: D'Alembert PDE

    Danny,

    No, I haven't seen that proved. Is that result useful for proving the other result? Or is it just that the one proof is analogous to the other?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41

    Re: D'Alembert PDE

    Ya, the proof is the same for both problems. Do you have (or can you get a copy of) Colton's book on PDEs (it a Dover book). In mine, it's on page 79.

    If not, I'll give you the details of this proof and let you provide details of the proof of your problem - Deal!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: D'Alembert PDE

    Thanks.... the library had the book Partial Differential Equations: An Introduction by David Colton (1988). Is that the correct text? But unfortunately the library's copy is not the Dover re-print. I looked at p79 but the theorem you cite is not there. Nor can I find it around that chapter.

    Would it be possible to tell me where it is by chapter/section instead of by page ? Thanks !
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41

    Re: D'Alembert PDE

    Ya - that's the book. I refer to section 2.4 "The initial boundary value problem for the wave equation in two indepedent variables"
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: D'Alembert PDE

    Oh man.... I just realized what was giving me trouble. Apparently I missed the obvious fact that \frac{d}{dt}u(x_0,t)=\left[\frac{\partial}{\partial t}u(x,t)\right]_{x=x_0}. I can't believe I didn't see that...

    Thanks for the help!
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,366
    Thanks
    41

    Re: D'Alembert PDE

    Yes, that's one of the secrets to proving your result.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. d'Alembert's formula
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: October 7th 2011, 11:02 AM
  2. PDE using D'Alembert's Solution
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: August 25th 2010, 04:35 AM
  3. d'alembert solution Help
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: May 16th 2010, 09:37 AM
  4. D'Alembert's ratio test
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 11th 2010, 12:01 AM
  5. D'Alembert's Solution
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: November 27th 2009, 05:54 PM

/mathhelpforum @mathhelpforum